What value of p makes this an evenly matched game of one-on-one, so that both players have an equal chance of winning before the coin is flipped?

Question

Now that LeBron James and Anthony Davis have restored the Los Angeles Lakers to glory with their recent victory in the NBA Finals, suppose they decide to play a game of sudden-death, one-on-one basketball. They’ll flip a coin to see which of them has first possession, and whoever makes the first basket wins the game.

Both players have a 50 percent chance of making any shot they take. However, Davis is the superior rebounder and will always rebound any shot that either of them misses. Every time Davis rebounds the ball, he dribbles back to the three-point line before attempting another shot.

Before each of Davis’s shot attempts, James has a probability p of stealing the ball and regaining possession before Davis can get the shot off. What value of p makes this an evenly matched game of one-on-one, so that both players have an equal chance of winning before the coin is flipped?

If you can post the complete solution — step by step — so that I can work through each step that would be greatly appreciated.

Answer 1

If we allow LeBron to steal even on the first possession, if Anthony wins the flip, we can figure as follows

Probability that LeBron wins on the first possession (this includes both LeBron winning by winning the flip and stealing) : .5 * .5 + .5 * p * .5 = .25 + .25p

Probability that Anthony wins on the first possession: .5 * (1-p) * .5 = .25 - .25p

After the first shot, each has equal probability on each shot given that they have possession, and Anthony always gets the rebound, so the probability that Anthony wins after the first shot, given that there is another shot, is 1 - p, and the probability that LeBron wins is p. There is .5 probability that there is no win on the first shot. Thus, using LeBron's probability,

.25 + .25p + .5p = .5

.25 + .75p = .5

.75 p = .25

p = 1/3

Note that, for Anthony, .25 - .25p + .5(1 - p) = .75 - .75p = .75 - .75(1/3) = .5

(If we did not allow LeBron to steal on the first possession, p = .5)